# Statistical Inference

Suppose we have a variable

$X$

(may or may not be a random variable) that represents the state of nature. We observe a variable $Y$

which is obtained by some model of the world $P_{Y|X}$

.Figure 2: Inference Setup

Suppose we know that

$X\sim \pi$

where $\pi$

is a probability distribution. If we observe $Y=y$

, then the *a posteriori*estimate of$X$

is given by Bayes Rule

$\text{Pr}\left\{X=x | Y=y\right\} = \frac{P_{Y|X}(y|x)\pi(x)}{\sum_{\tilde{x}}P_{Y|X}(y|\tilde{x})\pi(\tilde{x})} \propto P_{Y|X}(y|x)\pi(x).$

Since the estimate is only dependent on the model and the prior, we don’t actually need to compute the probabilities to figure out the most likely

$X$

.If we have no prior information on

$X$

, then we can assume $\pi$

is uniform, reducing Definition 77 to only optimize over the model.Since there are only two possible values of

$X$

in a binary test, there are two “hypotheses” that we have, and we want to accept the more likely one.With two possible hypotheses, there are two kinds of errors we can make.

Our goal is to create a decision rule

$\hat{X}: \mathcal{Y} \to \{0, 1\}$

that we can use to predict $X$

. Based on what the decision rule is used for, there will be requirements on how large the probability of Type I and Type II errors can be. We can formulate the search for a hypothesis test as an optimization. For some $\beta \in [0, 1]$

, we want to find

$\hat{X}_\beta(Y) = \text{argmin} \text{Pr}\left\{\hat{X}(Y)=0 | X=1\right\} \quad : \quad \text{Pr}\left\{\hat{X}(Y)=1|X=0\right\} \leq \beta. \qquad (1)$

Intuitively, our test should depend on

$p_{Y|X}(y|1)$

and $p_{Y|X}(y|0)$

since these quantities give us how likely we are to get our observations if we knew the ground truth. We can define a ratio that formally compares these two quantities.Notice that we can write MLE as a threshold on the likelihood ratio since if

$L(y) \geq 1$

, then we say $X=1$

, and vice versa. However, there is no particular reason that $1$

must always be the number at which we threshold our likelihood ratio, and so we can generalize this idea to form different forms of tests.MAP fits into the framework of a threshold test since we can write

$\hat{X}_{MAP} = \begin{cases} 1 & \text{ if } L(y) \geq \frac{\pi_0}{\pi_1}\\ 0 & \text{ if } L(y) < \frac{\pi_0}{\pi_1} \end{cases}$

It turns out that threshold tests are optimal with respect to solving Equation 1.

When

$L(y)$

is monotonically increasing or decreasing, we can make the decision rule even simpler since it can be turned into a threshold on $y$

. For example, if $L(y)$

is monotonically inreasing, then an optimal decision rule might be

$\hat{X}(y) = \begin{cases} 1 & \text{ if } y > c\\ 0 & \text{ if } y < c\\ \text{Bernoulli}(\gamma) & \text{ if } y = c. \end{cases}$

Last modified 11mo ago